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Suppose that $A$ acts on a finite group $G$ in such a manner that $C_G(V)=1$ and $C_G(\\alpha)$ has exponent $e$. We show that if $A\\cong S_4$ then the exponent of $G$ is $e$-bounded and if $A\\cong D_8$ then the exponent of the derived group $G'$ is $e$-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of auto"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1101.5367","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-01-27T19:23:54Z","cross_cats_sorted":[],"title_canon_sha256":"da85becea26372560265c7c3d9ba51b079e1720e4a3b930b622744bf52a2f876","abstract_canon_sha256":"61417d5c5f05a7e46b49a56369dccb2c11f1c9b3411b227df72b832ddb5d2064"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:30:35.616276Z","signature_b64":"dAklHGwcn7/VStVyM/NRMnMaNzSFFOWXomG49DCTJmfiAvIh1NzwhVb9cUFZjQSRSsq4yB2rWotNsTFUI/dNBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"767b13334e7070e8a22a4a8a12e7fa9109314de7184e77f21b4f060535aabb67","last_reissued_at":"2026-05-18T04:30:35.615694Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:30:35.615694Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the exponent of a finite group admitting a fixed-point-free four-group of automorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"E. Romano, P. Shumyatsky","submitted_at":"2011-01-27T19:23:54Z","abstract_excerpt":"Let $A$ be a group isomorphic with either $S_4$, the symmetric group on four symbols, or $D_8$, the dihedral group of order 8. Let $V$ be a normal four-subgroup of $A$ and $\\alpha$ an involution in $A\\setminus V$. Suppose that $A$ acts on a finite group $G$ in such a manner that $C_G(V)=1$ and $C_G(\\alpha)$ has exponent $e$. We show that if $A\\cong S_4$ then the exponent of $G$ is $e$-bounded and if $A\\cong D_8$ then the exponent of the derived group $G'$ is $e$-bounded. 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